Answer
$$\frac{6}{\sqrt{14}-6\sqrt{7}}=-\frac{6\sqrt{14}+36\sqrt{7}}{238}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{6}{\sqrt{14}-6\sqrt{7}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{6}{\sqrt{14}-6\sqrt{7}}\frac{\sqrt{14}+6\sqrt{7}}{\sqrt{14}+6\sqrt{7}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{6\sqrt{14}+36\sqrt{7}}{14+42\sqrt{2}-42\sqrt{2}-252} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{6\sqrt{14}+36\sqrt{7}}{-238} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{6\sqrt{14}+36\sqrt{7}}{238}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{14} + 6 \sqrt{7}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 6 } \cdot \left( \sqrt{14} + 6 \sqrt{7}\right) = \color{blue}{6} \cdot \sqrt{14}+\color{blue}{6} \cdot 6 \sqrt{7} = \\ = 6 \sqrt{14} + 36 \sqrt{7} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{14}- 6 \sqrt{7}\right) } \cdot \left( \sqrt{14} + 6 \sqrt{7}\right) = \color{blue}{ \sqrt{14}} \cdot \sqrt{14}+\color{blue}{ \sqrt{14}} \cdot 6 \sqrt{7}\color{blue}{- 6 \sqrt{7}} \cdot \sqrt{14}\color{blue}{- 6 \sqrt{7}} \cdot 6 \sqrt{7} = \\ = 14 + 42 \sqrt{2}- 42 \sqrt{2}-252 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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